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SPARSE PARTITION REGULARITY

Published online by Cambridge University Press:  13 October 2006

IMRE LEADER  and
PAUL A. RUSSELL
IMRE LEADER
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United KingdomI.Leader@dpmms.cam.ac.uk, P.A.Russell@dpmms.cam.ac.uk
PAUL A. RUSSELL
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United KingdomI.Leader@dpmms.cam.ac.uk, P.A.Russell@dpmms.cam.ac.uk
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Abstract

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.

Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

Keywords

05D10
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 93 , Issue 3 , November 2006 , pp. 545 - 569
Copyright
2006 London Mathematical Society

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